Find the roots of the quadratic equations by applying the quadratic formula.
$4x^2 - 4\sqrt3x + 3 = 0$

Given:

Given quadratic equation is $4x^2 - 4\sqrt3x + 3 = 0$

To do:

We have to find the roots of the given quadratic equation.

Solution:

$4x^2 - 4\sqrt3x + 3 = 0$

The above equation is of the form $ax^2 + bx + c = 0$, where $a = 4, b = -4\sqrt3$ and $c = 3$

Discriminant $\mathrm{D} =b^{2}-4 a c$

$=(4 \sqrt{3})^{2}-4 \times 4 \times 3$

$=48-48$

$=0$

$\mathrm{D}=0$

Let the roots of the equation are $\alpha$ and $\beta$

$\alpha=\frac{-b+\sqrt{\mathrm{D}}}{2 a}$

$=\frac{-4 \sqrt{3}+0}{8}$

$=\frac{-4 \sqrt{3}}{8}$

$=\frac{-\sqrt{3}}{2}$

$\beta=\frac{-b-\sqrt{\mathrm{D}}}{2 a}$

$=\frac{-4 \sqrt{3}-0}{8}$

$=\frac{-4 \sqrt{3}}{8}$

$=\frac{-\sqrt{3}}{2}$

Hence, the roots of the given quadratic equation are $\frac{-\sqrt{3}}{2}, \frac{-\sqrt{3}}{2}$.